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Splitting Methods for Partial Differential Equations with Rough Solutions
EMS Series of Lectures in Mathematics

Helge Holden (Norwegian University of Science and Technology, Trondheim, Norway)
Kenneth Hvistendahl Karlsen (University of Oslo, Norway)
Knut-Andreas Lie (University of Oslo, Norway)
Nils Henrik Risebro (University of Oslo, Norway)

Splitting Methods for Partial Differential Equations with Rough Solutions

Analysis and MATLAB programs

ISBN print 978-3-03719-078-4, ISBN online 978-3-03719-578-9
DOI 10.4171/078
April 2010, 234 pages, softcover, 17 x 24 cm.
36.00 Euro

Operator splitting (or the fractional steps method) is a very common tool to analyze nonlinear partial differential equations both numerically and analytically. By applying operator splitting to a complicated model one can often split it into simpler problems that can be analyzed separately. In this book one studies operator splitting for a family of nonlinear evolution equations, including hyperbolic conservation laws and degenerate convection-diffusion equations. Common for these equations is the prevalence of rough, or non-smooth, solutions, e.g., shocks.

Rigorous analysis is presented, showing that both semi-discrete and fully discrete splitting methods converge. For conservation laws, sharp error estimates are provided and for convection-diffusion equations one discusses a priori and a posteriori correction of entropy errors introduced by the splitting. Numerical methods include finite difference and finite volume methods as well as front tracking. The theory is illustrated by numerous examples. There is a dedicated web page that provides MATLAB codes for many of the examples.

The book is suitable for graduate students and researchers in pure and applied mathematics, physics, and engineering.

Keywords: Operator splitting, nonlinear partial differential equations, evolution equations, hyperbolic conservation laws, degenerate convection-diffusion equations, finite difference methods, finite volume methods, front tracking, Matlab codes

Further Information

Dedicated webpage for this book with MATLAB codes

Review in Zentralblatt MATH 1191.35005

Review in MR 2662342